AMC 8 2024 Problem 8
AMC8 2024 年第 8 题
基本计数原理★★☆☆☆
On Monday Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
周一那天，Taye有2美元。每天，他要么获得3美元，要么拥有的金额变成前一天的两倍。请问三天后，也就是周四，Taye 拥有的金额可能有多少种不同的结果?
- A.3
- B.4
- C.5
- D.6
- E.7
正确答案 · Correct Answer
D
解析 · Solution
解法1
第一天可能有多少美元值？只有2美元。第二天，你可以选择加3美元，或者翻倍，所以你可以有5美元或4美元。对于这些值中的每一个，你都有两个值。对于5美元，你可以有10美元或8美元；对于4美元，你可以有8美元或7美元。现在，对于这些值中的每一个，你都有两个值。对于10美元，你可以有13美元或20美元；对于8美元，你可以有16美元或11美元；对于8美元，你可以有16美元或11美元；对于7美元，你可以有14美元或10美元。最后一天，有11、11、16和16重复出现，最终剩下8 = 个不同的值。
继续这样操作，第二天得到7、8或10美元。从这里开始，唯一能让同一金额出现两次（乘以2或加3）的方法是如果或，但这两者都不成立。因此我们的答案是。Wijetunga 令表示Taye加3美元，表示Taye将钱翻倍。我们可以列出所有可能性： => => = 11， => => = 16， => => = 12， => => = 20， => => = 10， => => = 14， => => = 11， => => = 16。唯一的数值是10, 11, 12, 14, 16, 20。因此，我们要求的答案是
Solution 1
How many dollar values could be on the first day? Only 2 dollars. The second day, you can either add 3 dollars, or double, so you can have 5 dollars, or 4. For each of these values, you have 2 values for each. For 5 dollars, you have 10 dollars or 8, and for 4 dollars, you have 8 dollars or 7 dollars. Now, you have 2 values for each of these. For 10 dollars, you have 13 dollars or 20, for 8 dollars, you have 16 dollars or 11, for 8 dollars, you have 16 dollars or 11, and for 7 dollars, you have 14 dollars or 10. On the final day, there are 11, 11, 16, and 16 repeating, leaving you with 8 = different values. (minor formatting edits)
Continue as in to get 7, 8, or 10 dollars 2nd day. The only way to get the same dollar amount occurring twice (multiply or adding 3 ) from here is if or which both aren't true. Hence our answer is . Wijetunga Let represent Taye adding 3 dollars and represent Taye doubling his money. We can list out all the possibilities: => => = 11, => => = 16, => => = 12, => => = 20, => => = 10, => => = 14, => => = 11, => => = 16. The unique values are 10,11,12,14,16,20. Thus, our desired answer is
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